Optimal. Leaf size=49 \[ -i e^{2 i a} x^2-i e^{4 i a} \log \left (-x^2+e^{2 i a}\right )-\frac {i x^4}{4} \]
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Rubi [F] time = 0.03, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int x^3 \cot (a+i \log (x)) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int x^3 \cot (a+i \log (x)) \, dx &=\int x^3 \cot (a+i \log (x)) \, dx\\ \end {align*}
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Mathematica [B] time = 0.04, size = 137, normalized size = 2.80 \[ x^2 \sin (2 a)-i x^2 \cos (2 a)-\cos (4 a) \tan ^{-1}\left (\frac {\left (x^2-1\right ) \cos (a)}{x^2 (-\sin (a))-\sin (a)}\right )-i \sin (4 a) \tan ^{-1}\left (\frac {\left (x^2-1\right ) \cos (a)}{x^2 (-\sin (a))-\sin (a)}\right )-\frac {1}{2} i \cos (4 a) \log \left (-2 x^2 \cos (2 a)+x^4+1\right )+\frac {1}{2} \sin (4 a) \log \left (-2 x^2 \cos (2 a)+x^4+1\right )-\frac {i x^4}{4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 32, normalized size = 0.65 \[ -\frac {1}{4} i \, x^{4} - i \, x^{2} e^{\left (2 i \, a\right )} - i \, e^{\left (4 i \, a\right )} \log \left (x^{2} - e^{\left (2 i \, a\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.55, size = 50, normalized size = 1.02 \[ -\frac {1}{4} i \, x^{4} - i \, x^{2} e^{\left (2 i \, a\right )} + \frac {1}{2} \, \pi e^{\left (4 i \, a\right )} - i \, e^{\left (4 i \, a\right )} \log \left (x + e^{\left (i \, a\right )}\right ) - i \, e^{\left (4 i \, a\right )} \log \left (-x + e^{\left (i \, a\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 39, normalized size = 0.80 \[ -i {\mathrm e}^{2 i a} x^{2}-\frac {i x^{4}}{4}-i {\mathrm e}^{4 i a} \ln \left ({\mathrm e}^{2 i a}-x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 136, normalized size = 2.78 \[ -\frac {1}{4} i \, x^{4} - x^{2} {\left (i \, \cos \left (2 \, a\right ) - \sin \left (2 \, a\right )\right )} + \frac {1}{4} \, {\left (4 \, \cos \left (4 \, a\right ) + 4 i \, \sin \left (4 \, a\right )\right )} \arctan \left (\sin \relax (a), x + \cos \relax (a)\right ) - \frac {1}{4} \, {\left (4 \, \cos \left (4 \, a\right ) + 4 i \, \sin \left (4 \, a\right )\right )} \arctan \left (\sin \relax (a), x - \cos \relax (a)\right ) - \frac {1}{2} \, {\left (i \, \cos \left (4 \, a\right ) - \sin \left (4 \, a\right )\right )} \log \left (x^{2} + 2 \, x \cos \relax (a) + \cos \relax (a)^{2} + \sin \relax (a)^{2}\right ) - \frac {1}{2} \, {\left (i \, \cos \left (4 \, a\right ) - \sin \left (4 \, a\right )\right )} \log \left (x^{2} - 2 \, x \cos \relax (a) + \cos \relax (a)^{2} + \sin \relax (a)^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.22, size = 38, normalized size = 0.78 \[ -x^2\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,1{}\mathrm {i}-\ln \left (x^2-{\mathrm {e}}^{a\,2{}\mathrm {i}}\right )\,{\mathrm {e}}^{a\,4{}\mathrm {i}}\,1{}\mathrm {i}-\frac {x^4\,1{}\mathrm {i}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.22, size = 39, normalized size = 0.80 \[ - \frac {i x^{4}}{4} - i x^{2} e^{2 i a} - i e^{4 i a} \log {\left (x^{2} - e^{2 i a} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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